J. Phys. Chem. 1980, 84,975-979
975
Solvent Activity along a Saturation Line and Solubility of Hydrogen-Bonding Solids H. Buchowski," A. Ksiazczak, and S. Pietrzyk Department of Physical Chemistry, Technical University of Warsaw, 00-664 Warsaw, Poland (Received June 4, 1979)
Vapor pressure over saturated solutions, solvent activity (aB),and solubility of six phenols and benzoic acid in nonpolar solvents have been measured as functions of temperature. In (1- ug) is a linear function of reciprocal temperature; the slope is a product of the enthalpy of solution per mole of solute (h)and the quantity A, identified approximately as the average size of the multimer molecule. The values of h and h are nearly constant as functions of temperature. An expression relating the activity of the solvent and the mole fraction of solute in saturated solution, as well as a novel equation for the solubility curve in eutectic systems, has been derived and verified experimentally.
Introduction Solution-crystal equilibria have been extensively studied, but usually the measurements are confined to the determination of the solubility of a solid. If both components are completely miscible in the liquid state but completely immiscible in the solid state, the solubility curve of a solid A at constant pressure is expressed by eq 1, where XAsat and y ~ * are * ~ the mole fraction and the ac-
tivity coefficient of A in a saturated solution respectively; a hypothetical liquid A supercooled to a temperature T , lower than melting temperature T,, is taken as a reference standard state; A,HfA is the enthalpy of fusion at T; and R is the gas constant. The integral in eq 1 depends on the properties of A only and defines the so-called ideal solubility. Various explicit formulas for ideal solubility have been proposed; with the assumption of A,HfA being independent of temperature, integration gives the wellknown Schroder equation.2 More accurate expressions based on less restrictive assumptions regarding the variation of enthalpy of fusion with temperature have been discussed by Hildebrand3 and recently by Gokcena4 The second term on the right-hand side of eq 1 is the activity coefficient in saturated solution; it depends on properties of both solute and solvent, on composition of the solution, and on temperature. But little is known about this function. Hildebrand's equation, based on solubility parameter theory, is the best expression for yAmt of nonpolar solutes in nonpolar solvents at temperatures not very distant from 25 "C. A method for the calculation of the activity of polar solutes from solubility data has been proposed by Kohler and c o - ~ o r k e r s . ~ Solubility of an associating solid in nonpolar solvents depends strongly on the activity coefficient in saturated solution (yAsat),and the solubility curve deviates much from the ideal one. Hildebrand's solubility parameter equation is not applicable to hydrogen-bond-forming solutes. Therefore, a search for an equation suitable for associated systems is needed. In discussing properties of solutions of compounds capable of self-association, one should take account of the existence of multimers, dimers, trimers, etc. The effective mole fraction of solute, Le., the sum of true mole fractions of all multimers, Ex;, is an important parameter characterizing self-association. According to the theory of ideal associated solutions, Cxi = 1 - UB, where UB is the activity of solvent B. The last equation invokes the need for studying the activity of the solvent in saturated solution. To determine solvent ac0022-3654/80/2084-0975$0 1 .OO/O
tivity we measured the vapor pressure over saturated solutions. The solutions of six phenols and of benzoic acid in nonpolar solvents have been investigated. The method of investigation and results obtained are reported and discussed in the rest of the paper.
Experimental Section 1. Materials. Purification methods and characteristics of purified 2,3-, 3,4-, and 2,6-dimethylphenol, 2,6-di-tertbutyl-4-methylphenol,2-nitro-5-methylpheno1,and benzoic acid, as well as of the solvents, are described in previous papers.61' 4-tert-Butylphenol (Koch-Light pure product) was purified by repeated vacuum sublimation (melted at 100.15 O C ) and its purity, determined by vapor-phase chromatography, was 99.6%. 2. Apparatus. Vapor pressure was measured in the apparatus shown in Figure 1. Three connected glass tubes (J1,J,, J3)partly filled with mercury are the main parts of the apparatus. In the side tubes J1and J2,the samples (1-5 cm3)of the solution and of the solvent are placed. The middle tube, J3,is connected with three lines: pressure, evacuation, and high-vacuum. The pressure line includes a compressed nitrogen cylinder, the pressure-adjusting device R, the column M containing type 4A molecular sieves, and a cutoff tap K3. The evacuating line consists of a vacuum pump, a molecular-sieves-filledcolumn, a flow regulator, and a cutoff tap K4. The pressure and evacuating lines are used to compress and expand vapors in tubes J1 and Jz. The high-vacuum line consists of a vacuum pump, a vacuum gauge G, and a cutoff tap Kl; it allows for a vacuum of the order of 1 Pa in the tube J3; thus, absolute vapor pressure measurements are possible. The temperature of the thermostating jacket surrounding the tubes is held constant to 0.002 K. Mixing of liquids is achieved by vibrating the rod to which the tubes and the jacket are attached. 3. Procedure. Pure solvent is introduced into J1,and its mixture with a solid in 1:5 molar proportions is placed in tube J2; a great excess of the solid facilitates fast equilibration of the mixture. The temperature is elevated so that the solid phase disappears, and the sample is degassed in a series of expansion-compression runs as described elsewhere.' After the sample is degassed, the mercury is held at a high level while the temperature is gradually lowered. This causes crystallization of the solid in the upper part of tube J2. Tube J3is then opened to high vacuum, and the pressure is lowered until it reaches the value of mercury vapor pressure. The differences in mercury levels between tubes J1and J3(ah,,)and between J1 and J2 (Ah,,) are measured to 0.01 mm with a cathetometer. The Ah13 value is used to determine the vapor 0 1980 American Chemlcal Society
976
Buchowski et al.
The Journal of Physical Chemistry, Vol. 84, No. 9, 1980
pressure of the solvent pfB. Application of Antoine’s equation to the pfBdata permits the calculation of the temperature in the apparatus. The difference between the temperature measured in the thermostatic jacket and the calculated one never exceeded 0.03 K. The difference, Ap, between the solvent vapor pressure and that of the solution is obtained from Ahl2. Since the vapor pressure of solids under consideration is negligibly small, it is assumed that the vapor pressure over the solution is equal to the partial pressure of the solvent.8 Hence, it follows
to pump ..c 1 I
Jzl I
I I J
-D
Y!-. The solubility was determined as a function of temperature by the synthetic method in separate experiments. The pressure effect on solubility was neglected.
-+to nitrogen
Flgure 1. Diagram of the apparatus.
c In(l-a,)
Theoretical Section We consider a binary system composed of a vapor phase in equilibrium with a liquid solution and a solid phase formed by pure A. The two-component, three-phase system has one degree of freedom, so the activity of the solvent (aB) is an implicit function of temperature only. According to the general formula for the derivative of an implicit function? d In (1- a B )
to pump
-1
’L
d In (1 - a B )
\
9
-. By putting into eq 3 first y = plA - pfAB and second y = xA, we calculate two thermal coefficients of In (1- aB): one along the saturation line denoted by an index “sat” and the second for constant mole fraction of the solute A. By subtracting the derivative for y = xA from the first one, we eliminate the derivative at constant uA. d In (1 - aBlsat = ( d In (1 - a B ) ) T [ aT-1 d In aA
In aASat 87-1
-4 -
b
Flgure 2. Activity of the solvent (a,) in saturated solution as a function of temperature ( T ) . T,: melting temperature. Solutes: (1) 2,6-diferf-butyC4-methylphnol; (2) 2,Mimethylphenol; (3) 2,Wimethylphenol; (4) 3,4dimethylphenol, Solvents: (I) benzene; (11) cyclohexane. (0) calculated from vapor pressure (eq 2); (0)from solubility (eq 16).
The first derivative in braces, being the differential form of eq 1, is equal to the enthalpy of fusion of A
With the identity (1- u B ) d In (1- u B ) = -uB d In
UB
(5)
we transform the last term of the right-hand side of eq 4 into d In aB
d In (1 - aB)
(6)
-
d In aAsat dT’
= A,Hf,/R
The second term is related to the enthalpy of mixing, HE
(
~ ( x In A u;;~xB
In ag) )xA
= (d(@K,,)
= XA
HE/R (10)
Putting eq 5 into the Gibbs-Duhem equation,1° we obtain Finally dlnaA From eq 4, 6, and 7 we obtain
- d In (1- adsat d P -d In aAsat
h(
dT1
(7)
where X is defined by eq 7 and
h R = A m p A + HE/xABat ~ ( x A In
+$(
aA + X B In ag) dT1
(12)
Results 1. Variation of (1 - aB)satwith Temperature. In Figure 2 and 3 the logarithm of (1- aB) for saturated solutions
The Journal of Physical Chemistry, Vol. 84, No. 9. 1980
Solvent Activity along a Saturation Line
0
04
02
0.6(1-1).703
T T , Flgure 3. The same as in Figure 2. Solutes: (5) 2-nitro-5-methylphenol; (6) 4-tert-butylphenol; (7) benzoic acid. Solvents: (111) ethyl acetate; (IV) carbon tetrachloride.
of phenols in cyclohexane and benzene are plotted against 1/T. Experimental points (open circles) for all systems lie on the straight lines. I t follows from eq 11 that along the saturation curve Ah is constant within the limits of the precision of the measurements. Integration of eq 11 from T, to T gives eq 13. The constant value of the product - In (1 - u B ) = ~ Xh(T1 ~ ~ -
7',-')
(13)
Ah may be due to a compensation of changes of both X and h. But the coefficient h can be found separately. Equations 7 and 2 give
where p = pfB- Ap is the vapor pressure over a saturated solution. In Figure 4 the Xsat calculated by eq 14 is shown as a function of temperature. The points are scattered, but nevertheless it can be concluded that the coefficient X exhibits no marked variation with temperature. The X can be taken as a constant within experimental error. We estimate the relative error of X at about 5%, taking into account the precision of vapor pressure and solubility measurements.l' The error limits (f0.05X) are marked in Figure 4. 2. Solubility Curve. Equation 7 can be transformed as follows: -1
X(1 - xA)/xA = aB/(1 - ag) = (1
(15)
Hence, - In (1 - UB)
= h [1
+ X(1 - xA)/xA]
(16)
Equations 13 and 16 give the solubility curve: In [1 + h(1 - XA)/XA]sat
Xh(T' -
!#?m-')
(17)
Experimental verification is done in two ways. First, the left-hand side of eq 17 is calculated from the solubility data using X values found previously (see Figure 4) and plotted as a function of T1in Figures 2 and 3 (full points). As expected the points follow the same straight line as does the value of - In (1 Second, the saturation temperature is plotted as a function of mole fraction of the
Figure 4. Variation of of temperature.
977
(see eq 7) for saturated solution as a function
solute in Figure 5. Experimental data are represented by points; the curves are calculated with eq 17 written in the form:
T1= Tm-l+ (Ah)-' In [I + X ( l
- X A ) / ~ A ] ~(18) ~ ~
Constants X and h are adjusted by a nonlinear optimization method.12 The experimental points and the calculated curves agree very well.
Discussion According to eq 17 two factors determine the solubility of a solid A in a solvent B. The first, h, the enthalpic factor defined by eq 12, can be transformed as follows: hR = P A ' - HAs + RA'- HfL + (xB/xA)(RB' - Hfd) = Rk - HAs
+ (xg/xA)(RB' - Hfd)
(19)
= AsolnH where HfA1and HAB are molar enthalpies at temperature T of the supercooled liquid A and of the solid_A, respectively, Hfd is the enthalpy of liquid B, and HA' and H& are partial molar enthalpies in saturated solution. As0hH is the enthalpy of solution per mole of the solute A. In Table I the values of h are listed and compared with AmHfA(Tm)/R, i.e., the enthalpy of fusion at melting temperature divided by the gas constant. This enthalpy contributes more than 80% of the h value. The apparent constancy of h is most likely due to the compensation of variations of A m p A and HE/xASatwith temperature. The second factor, A, defined by eq 7, is the derivative of In (1 - uB) with respect to In UA at constant temperature. In an ideal solution, 1 - U B = 1 - XB = XA = uA, SO X identically equals 1. Thus X in eq 1,13, or 17 is a measure of nonideality of the saturated solution. If deviations from ideality are mainly due to association, h can be interpreted in terms of the theory of ideal associated solutions. In this case 1 - UB = E x , = CKlixli and UA = xl, where x1 and xi are the mole fractions of monomers or i-mers, respectively, and the association constant Kli = x i x l i . Hence it follows that d In Exi A= d In x 1
x
= (xl/Cxi)CiKljxli-l=
Cix,/Cxi = X
(20)
where X is the mean association number, Le., the mean value of the number of monomolecules per multimer
978
The Journal of Physical Chemistry, Vol. 84,
No. 9,
Buchowski et ai.
1980
I
I
I
I
0,4
02
I
I
I
I
0.6
I
x*
*,@
Figure 5. Solubility curves. Designation of systems same as in Figures 2 and 3 or in Table I. Points: experimental results: curves: calculated by eq 18.
TABLE I: Constants of Equations 13 and 17 (Ah )vP 3 103 ~b
no.
system solute (A)-solvent (B)
T,, Ku
1, I
2,6-di-tert-butyl-4-methylphenolbenzene 2,6-di-tert-butyl-4-methylphenolcyclohexane 2,6-dimethylphenol-benzene 2,6-dimethylphenol-cyclohexane 2,3-dimethylphenol-benzene 3,4-dimethylphenol-benzene 3,4-dimethylphenol-cyclohexane 2-nitro-5-methylphenol-benzene 2-nitro-5-methylphenol-carbon tetrachloride 2-nitro-5-methylphenol-ethylacetate 4-tert-butylphenol-benzene benzoic acid-benzene
343.3
1,I1 2,I 2,I1 3,I 4,I 4,I1 5,I 5,IV 5,I11 6,I 7,I
hs
mean rmsdh mean rmsdh
9
lo3
hvPC hQd
Kd
8T, Ke
h"P;
10 Kf
AmHAf/R,
2.30
1.83 1.83 2.39 1.58 1.58 2.50 2.50
l o 3 Kg
1.79 0.02 2.25 0.03
318.77
3.11 8.31 345.72 4.10 338.26 4.98 14.77 327.8 3.67 4.95 373.3 394.9
0.11 1.35 0.08 1.38 0.26 5.65 0.11 1.50 0.05 1.55 0.13 2.16 0.13 3.12 0.57 11.9 0.07 1.49 0.03 1.75 0.08 1.99 0.07 2.55
3.15 0.09 1.23 3.29 0.06 1.66 4.14 0.05 2.12
2.28 1.88 2.69 1.95 1.62 2.32 2.24
0.1 2.74 0.4 2.31 0.4 0.1 2.46 0.1 2.49
0.04 1.40 2.40 0.5 2.57 0.05 1.70 3.35 0.4 1.98 0.06 1.69 2.24 1.1 1.96
2.50 2.08
T , is the melting temperature of the solute. ( a h b = - In (1- a g ) / ( T - ' - T&') in saturated solution. hVp = X A ~ B / X B( ~a g ) ; a g is the activity of the solvent and X A (= 1 - x g ) is the mole fraction of the solute, both in saturated h, and h , are the constants of eq 20 calculated from the solubility data alone. e 6 T is the average deviation besolution. . is the enthalpy of fusion of the soltween the measured temperature and the calculated one, f h, = ( ~ h ) ~ / h g, ~arnHAf ute A at melting temperature. Mean = y = x i = l n y J n ;rmsd = root mean square deviation = [ C i , l n ( y j -"7)2/n]1'2.
molecule. The values of X of phenols (see Table I) agree fairly well with the above interpretation. The X's are higher in cyclohexane than in benzene and decrease in the order of increasing steric effect of alkyls on an OH group:7 3,4- > 2,3- > 2,6-dimethylphenolqIdentification of X with the mean association number enables the explanation of its constancy along the saturation line: X is a function of association constants and of mole fraction xAsat; as the temperature is elevated, association constants decrease and tend to diminish A, but solubility increases with temperature and tends to enlarge A. These two effects tend to cancel each other. The solubility of benzoic acid in benzene offers further arguments in favor of the proposed interpretation of X. The solution of benzoic acid in benzene, because of
structural similarity of solute and solvent, is expected to be close to an ideal associated one. Since benzoic acid exists in solution mainly in the form of cyclic dimers,13 at xA > 0.05, X approaches 2. The curves XA(!~') for constant X values have been calculated by eq 21, where K(T) is the x A ( T ) = X(X - 1)[K(T)(2- X)2 + (X - 1)2]-1(21) dimerization constant of the benzoic acid in benzene solution, expressed in mole fraction units. Variation of K as a function of temperature has been calculated according to the van't Hoff isobar; the standard enthalpy, AHOlR = 3924 K, and standard entropy of dimerization, AS"f R = -4.366, have been calculated from the IR spectroscopic data given by Allen, Watkinson, and Webb.14 As can be seen in Figure 6, the curve calculated for X = 1.940 agrees
J. Phys. Chem. 1980, 84, 979-980 t/'C
I
979
framework of Project PR 03.10.1.
I
References and Notes
0
"
"
0;z
"
"
'
0.4
06
08
(1) I. Prigogine and R. Defay, translated by D. H. Everett, "Chemical Thermodynamics", Longmans, Green and Co., New York, 1954. (2) I. Schroder, Z. Phys. Chem., Stoechiom. Verwandschaftsl., 11, 449 (1893). (3) J. H. Hildebrand and R. L. Scott, "Solubility of Nonelectrolytes", 3rd ed., Reinhold, New York, 1950. (4) N. A. Gokcen, J. Chem. Soc., Faraday Trans. 1, 438 (1973). (5) F. Kohler, G. H. Findenegg, and M. Bobik, J. Phys. Chem. 78, 1709 (1974); E. Libermann and F. Khler, hhnatsh. Chem.,9% 2514 (1968). (6) H. Buchowski, U. Domaiska, A. Ksiazczak, and A. Maczytkki, Rocz. Chem., 49, 1889 (1975). (7) H. Buchowski and A. Ksiazczak, Rocz. Chem., 48, 65 (1974); 50, 1755, 1965 (1976). (8) I f the vapor pressure of a solid is too large to be neglected, it is subtracted from the measured value of Ap. (9) H. Margenau and G. H. Murphy, "The Mathematics of Physlcs and Chemistry", 2nd ed., Van Nostrand-Reinhold, Princeton, NJ, 1956. (10) The small effect of pressure on activities in condensed phases has been neglected. (1 1) Vapor pressure over saturated solution and solubility are determined in separate experiments, and xALlst corresponding to a given temperature Is found from the smoothed solubility curve. If the curve tmt - xmtis flat, the error due to interpolation is large; S x l x (1 x) for solutions of phenols in cyclohexane is greater than 20% and determination of A is impossible. (12) H. H. Rosenbrock, Comput. J., 3, 175 (1960). (13) H. Buchowski, J. Phys. Chem., 73, 3520 (1969). (14) G. Allen, J. G. Watkinson, and K. H. Webb, Spectrochim. Acta, 22, 807 (1966). (15) F. S. Mortimer, J . Am. Chem. Soc., 45, 633 (1923); J. Chipman, ibid., 46, 2445 (1924).
xA
Flgure 6. Solubility curve of benzoic acid in benzene calculated from spectroscopic data (see text). Points: data by Mortimer and by ChIpman.l5
very well with the solubility curve calculated from the data given by Mortimer and by Chipman.15
Acknowledgment. The authors are grateful. for the support of this work provided by the Institute of Physical Chemistry, Polish Academy of Sciences, within the
Determination of the Critical Exponents in Liquid Ternary Mixtures by a Nonlinear Dielectric Method J. ZioYo," 2. Ziejewska, and J. Piotrowska Institute of Physics, Silesian University, Uniwersytecka 4, 40-007 Katowice, Poland (Received August 13, 1979) Publication costs assisted by the Polish Academy of Sciences
Measurements of dependence Ae/E2 (where A€ = €high field - tiow field and E is the electric field strength)~vs.T T, (T, is the critical solution temperature of the mixture at the critical point and T is that outside it) are presented. At the critical point A t / E 2 1/(T - TJf is valid, whereas outside the critical point the following is valid: A t / E 2 l / ( T - T*)fwhere T* C T,, One can state, using this method, that the mixture of the critical solutions of nitrobenzene-hexane and nitroethane-hexane behaves as a critical solution.
-
-
-
Introduction The nonlinear dielectric effect (NDE) is defined by the quantity A t / E 2 (where Ae = th' h field - tlowfield and E is the electric field strength). This henomenon is used for investigations of intermolecular and intramolecular interactions in liquid dielectrics. It is also used for investigations of pretransitional effects near the phase transition, because this effect is extremely sensitive to fluctuations of the dielectric permittivity of a 1iquid.l Though the first work concerning critical solutions was carried out by Piekara2 in 1936 greater interest in using NDE for investigations of phase transition was not observed until several years later.3-6 All hitherto existing NDE investigations were limited to measurements a t the critical point. The second-order transition that is observed at a solution critical point is similar to a critical gas-liquid phase transition. For this case, near the temperature of phase separation (T,) the following dependence is ~ a l i d : ~ - ~ At E2
1 N
( T - TJS
(1)
0022-3654/80/2084-0979$01 .OO/O
where f is the critical index, dependent on the difference in dielectric permittivities of components of the The first-order transition, similar to the liquid-gas transition, ought to be observed outside the critical solution point. The following question arises: Is NDE a method sensitive to the difference between first-order and secondorder transitions? In order to answer this question NDE measurements have been made for critical and noncritical concentrations.
Results The subjects of our measurements were solutions of nitrobenzene in hexane (NB-H) and nitroethane in hexane (NE-H). Coexistence curves for NB-H (open circles) and NE-H (closed circles) are presented in Figure 1. Figure 2 shows results of measurements of (At/E2)-lvs. T - T, for different NB-H solution concentrations (critical solution: 0.43 (crosses); and noncritical solutions: 0.35 (closed circles), 0.5 (open circles)). A qualitative difference between results 0 1980 American Chemical Society
